that at least one pair of samples differs.  Let’s say you are interested in treatments A, B and C in an experiment, and will conduct hypothesis tests for differences between A and B, A and C, and B and C, each with a significance level of alpha = 0.05.

Each time you do an independent test, the individual Type I error is 0.05, so the probability of not making an error is 0.95.  The probability of not making such an error on three consecutive tests is 0.95 * 0.95 * 0.95 = 0.8574.  So the Type I error rate overall, or the family-wise error rate, is 1-0.8574 = 0.1426.  This is considerably more than the individual rate, and it increases as the number of comparisons increase.

If a is the probability of individual comparison-wise type I error, then the probability a_FW of family-wise type I error is usually calculated as follows:

where C is the total number of individual comparisons made.  You can see that to keep the overall family-wise error rate at 0.05 you will need to substantially reduce the alpha level for each test.